Question

The 1st, 5th, and 13th terms of an arithmetic sequence are the first three terms of a geometric sequence with a common ratio of 2. If the 21st term of the arithmetic sequence is 72, calculate the sum of the first 10 terms of the geometric sequence.

Answer 1

the 1st term of the A.S. is a, and the 5th term is a + 4k

from the G.S. with a common ratio of 2
(a + 4k)/a = 2
a + 4k = 2a
a = 4k

then from the 21st term of the A.S.
a + 20k = 72
substitute a = 4k
4k + 20k = 72
24k = 72
k = 3

a = 4*3
a = 12

the sum of the first 10 terms of the G.S. is
S_10 = 12*(1 - 2^10)/(1 - 2)
S_10 = 12276